Square root of a multivector in 3D Clifford algebras
Adolfas Dargys
Semiconductor Physics Institute
Artūras Acus
Vilnius University
Published 2020-03-02


geometric Clifford algebra
experimental mathematics
square root of multivector
infinitely many roots
Riccati equation

How to Cite

Dargys A. and Acus A. (2020) “Square root of a multivector in 3D Clifford algebras”, Nonlinear Analysis: Modelling and Control, 25(2), pp. 301–320. doi: 10.15388/namc.2020.25.16519.


The problem of square root of multivector (MV) in real 3D (n = 3) Clifford algebras Cl3;0, Cl2;1, Cl1;2 and Cl0;3 is considered. It is shown that the square root of general 3D MV can be extracted in radicals. Also, the article presents basis-free roots of MV grades such as scalars, vectors, bivectors, pseudoscalars and their combinations, which may be useful in applied Clifford algebras. It is shown that in mentioned Clifford algebras, there appear isolated square roots and continuum of roots on hypersurfaces (infinitely many roots). Possible numerical methods to extract square root from the MV are discussed too. As an illustration, the Riccati equation formulated in terms of Clifford algebra is solved. 

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